Download x and (−x) = x . If y is an odd power of x, as in y=x or y=x , it is an odd

College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
x and (−x) = x . If y is an odd power of x, as in y = x or y = x , it is
an odd function of x (because (−x) = x and (−x) = x ).
The graph of an even function is symmetric about the y-axis. Since
f(−x) = f(x) , a point (x, y) lies on the graph if and only if the point (-x,y)
lies on the graph (Figure below). A reflection across the y-axis leaves the
graph unchanged.
The graph of an odd function is symmetric about the origin. Since
f(−x) = −f(x), a point (x, y) lies on the graph if and only if the point(-x,y) lies on the graph (Figure below).
Equivalently, a graph is symmetric about the origin if a rotation of 180°
about the origin leaves the graph unchanged. Notice that the definitions
imply both x and -x must be in the domain of ƒ.
Ex:- Say whether each of the following functions is even, odd, or neither.
Give reasons for your answer.
1. f(x) = 3 2. f(x) = x
solution:1.
3. f(x) = x + 3x − 1
2.
3.
( )=
+
−
= (− ) + (− ) −
25
= (− ), thus ( ) is even.
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
1.5 Combining Functions; Shifting and Scaling Graphs
In this section we look at the main ways functions are combined or
transformed to form new functions.
Sums, Differences, Products, and Quotients
Like numbers, functions can be added, subtracted, multiplied, and divided
(except where the denominator is zero) to produce new functions. If ƒ and
g are functions, then for every x that belongs to the domains of both ƒ and
g (that is, for ( ∈ ( ) ∩ ( )) ), we define functions
and
by the formulas
+ , −
Notice that the sign + on the left-hand side of the first equation represents
the operation of addition of functions, whereas the + on the right-hand side
of the equation means addition of the real numbers ƒ(x) and g(x).
At any point of
( ) ∩ ( ) at which
function by the formula
( ) ≠ 0, we can also define the
Functions can also be multiplied by constants: If c is a real number, then
the function cƒ is defined for all x in the domain of ƒ by
Ex- Find the domains and ranges of ,
and
= √ − 1.
Solution:-
,
+
and .
where
=
.
The graph of the function f + g is obtained from the graphs of ƒ and g by
adding the corresponding y-coordinates ƒ(x) and g(x) at each point
x ∈ D(f) ∩ D(g), as in the following figure:26
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
Composite Functions
Composition is another method for combining functions.
The definition says that
° can be formed when the range of g lies in the
domain of ƒ. To find ( ° )( ), first find g(x) and second find ƒ(g(x)). The
following two figures (a and b) pictures
°
as a machine diagram and
shows the composite as an arrow diagram respectively.
(a)
(b)
27
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
Ex:-
Solution:-
Shifting a Graph of a Function
To shift the graph of a function
= f(x) straight up, add a positive
constant to the right-hand
hand side of the formula
of a function
= f(x). To shift the graph
= f(x) straight down, add a negative constant to the right-
hand side of the formula straight down. To shift the graph of
the left, add a positive constant to x. To shift the graph of
right, add a negative constant to x.
Ex:-
Solution:-
28
= f(x) to
= f(x) to the
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
Scaling and Reflecting a Graph of a Function
To scale the graph of a function y = f(x) is to stretch or compress
it, vertically or horizontally. This is accomplished by multiplying the function
ƒ, or the independent variable x, by an appropriate constant c. Reflections
across the coordinate axes are special cases where c = -1.
Ex:- Give an equation for the stretched or compressed graph of the following functions:-
Solution:-
1.6 Trigonometric Functions
This section reviews the basic trigonometric functions. The
trigonometric functions are important because they are periodic, or
repeating, and therefore model many naturally occurring periodic
processes.
Radian Measure
In navigation and astronomy, angles are measured in degrees, but in
calculus it is best to use units called radians because of the way they
simplify later calculations. The radian measure of the angle ACB at the
29
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
center of the unit circle (The following figure) equals the length of the arc
that ACB cuts from the unit circle.
Figure above shows that s = rθ is the length of arc cut from a circle of
radius r when the subtending angle producing the arc is measured in
radians. Since the circumference of the circle is 2π and one complete
revolution of a circle is 360°, the relation between radians and degrees is
given by
π radians = 180°.
for example, 90° in radian measure is
π
π
90
= rad,
180
2
and 3π/2 radians is
3π 180
.
= 270°
2 π
An angle in the xy-plane is said to be in standard position if its vertex
lies at the origin and its initial ray lies along the positive x-axis (Figure
below). Angles measured counterclockwise from the positive x-axis are
assigned positive measures; angles measured clockwise are assigned
negative measures.
30
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
When angles are used to describe counterclockwise rotations, our
measurements can go arbitrarily far beyond 2π radians or 360°. Similarly,
angles describing clockwise rotations can have negative measures of all
sizes (Figure below).
From now on in these lectures it is assumed that all angles are measured
in radians unless degrees or some other unit is stated explicitly. When you
do calculus, keep your calculator in radian mode.
The Six Basic Trigonometric Functions
You are probably familiar with defining the trigonometric functions of
an acute angle in terms of the sides of a right triangle (Figure below).
We extend this definition to obtuse and negative angles by first placing the
angle in standard position in a circle of radius r. We then define the
trigonometric functions in terms of the coordinates of the point P(x, y)
where the angle’s terminal ray intersects the circle (The following figure).
31
College of Engineering / 1st Class Mathematics Lectures Lecturer: Mr. Ali Nu'man Ibrahim
Periodicity and Graphs of the Trigonometric
Trigonometric Functions
The
he six basic trigonometric functions are periodic,
periodic hence
………..and so on.
When we graph trigonometric functions in the coordinate plane, we usually
denote the independent variable by x instead of
. See Figure below
As we can see in Figure 1.73, the tangent and cotangent functions have period
32